Proof step e-435b07172be58365

preserver_image affine, shift 0 → concludes about Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 1. JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): Polynomial.Splits.comp_of_degree_le_one

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"-2\",\"intercept\":\"3\"}"
}

Premises

• Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "lower": { "closed": true, "value": "3/2" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 2657c5bf20530609ca2dcd6d22f64c5ba0a2eb91c76cb0260dec8145d6e7d769, 10,420 bytes (2,926 compressed); download JSON

Cited by

• n-25dfcdb2c8790bf0 (one_signed_coefficients)
• n-25dfcdb2c8790bf0 (interlacing)
• n-25dfcdb2c8790bf0 (real_rooted)